14

This question is inspired by and/or a generalization of this question about the "reciprocal addition" operation.

Consider the following:

  • One is tempted to say multiplication is simply "addition under the logarithm", in the sense that $z = x y$ if and only if $\log z = \log x + \log y$ (for positive reals, anyway).

  • Convolution is simply pointwise multiplication under the Fourier transform: $h = f*g$ iff $\hat h = \hat f\hat g$.

  • Relativistic velocity addition is just regular addition under the map $\tanh^{-1}(\cdot/c)$.

  • "Reciprocal addition", $(x,y)\mapsto\cfrac1{\frac1x+\frac1y}$, is just addition under the reciprocal.

  • In general, given a binary operation $\oplus$ and a bijection $f$, one can construct a new operation "$\oplus$ under $f$" as $(x,y)\mapsto f^{-1}(f(x)\oplus f(y))$.

Commutativity and associativity are automatically inherited from the original operation, and the identity element is the inverse image of the original identity (if it exists). Of course, one can easily generalize this to $n$-ary operations. My question is twofold:

  1. Does this idea of "performing an operation under the action of a bijection" have a standard name, perhaps in some specialized context? Can it be considered a group-theoretic conjugation? A differential-geometric pullback? Some category-theoretic operation?

  2. What would be a good thing to call it in general? Ideally one would like to be able to say "this operation is just multiplication blah blah the logarithm, so naturally it's commutative and associative and the identity is $e$" in a way that people would be likely to get what you mean.

Community
  • 1
  • In regards to 2, are you looking for something that generalizes to (set, operation) pairs which are not explicitly groups? If not in particular, you'd simply call $f$ a group isomorphism. – Axoren May 15 '16 at 05:08
  • 1
    Yes, no group structure is necessary. Just two sets $X$ and $Y$ with a bijection $f:X\to Y$ and an operation $Y^n\to Y$. –  May 15 '16 at 16:29

2 Answers2

6

These are all instances of conjugation. The operation $(x,y)\mapsto f^{-1}(f(x)\oplus f(y))$ is a conjugate of the operation $(u,v) \mapsto u \oplus v$.

Michael Hardy
  • 1
4

There are probably several names around, but for me it is called "the operation obtained by transport of structure". A nice example is the one I wrote down in this answer.

Anne Bauval
  • 34,650
Johannes Huisman
  • 4,034
  • 12
  • 21